Direct Product Tables

These tables contain the irreducible representations spanned by all possible direct products in the given point groups.  For example in O symmetry the direct product E x T1 spans (contains) the irreducible representations T1+ T2.   All tables are symmetric about the diagonal. Click here for the tables in PDF format (suitable for printing).

For the Point Groups O and Td (and Oh)

  A1 A2 E T1 T2
A1 A1 A2 E T1 T2
A2   A1 E T2 T1
E     A1+ A2+ E T1+ T2 T1+ T2
T1       A1+ E + T1+ T2 A2+ E + T1+ T2
T2         A1+ E + T1+ T2

For the Point Groups D4, C4v, D2d (and D4h = D4 + Ci)

  A1 A2 B1 B2 E

A1

 A1 A2 B1 B2 E

A2

   A1 B2 B1 E

B1

     A1 A2 E

B2

       A1 E

E

         A1+ A2+ B1+ B2

For the Point Groups D3 and C3v

  A1 A2 E
A1 A1 A2 E
A2   A1 E

E

     A1+  A2+ E

For the Point Groups D6, C6v and D3h*

  A1 A2 B1 B2 E1 E2
A1 A1 A2 B1 B2 E1 E2
A2   A1 B2 B1 E1 E2
B1     A1 A2 E2 E1
B2       A1 E2 E1
E1         A1+ A2+ E2 B1+ B2+ E1
E2           A1+ A2+ E2

* in D3h make the following changes in the above table

In Table In D3h
A1 A1'
A2 A2'
B1 A1''
B2 A2''
E1 E''
E2 E'